Aid is help — foreign aid, first aid — and also the verb, to aid. An aide is a person who assists someone, typically an official or politician. The E makes it a person, borrowed from French.
The core difference
Help itself, and a person who provides it.
- aid — assistance, or to assist: humanitarian aid, aid the investigation.
- aide — an assistant: a senior aide, a nurse's aide.
The hyphenated forms
Aide-de-camp is a military officer assisting a senior commander, and is where the E comes from — French. A hearing aid takes no E, because it is a device rather than a person. That is the reliable test: people take the E.
The common slip
"A political aid" describes assistance to a party rather than a person working for one. It is a small error but it reaches print regularly.